Aqueous Redox Chemistry and the Electronic Band Structure of Liquid

Nov 6, 2012 - The effect is, however, badly exaggerated by the generalized gradient approximation leading to systematic underestimation of redox poten...
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Aqueous Redox Chemistry and the Electronic Band Structure of Liquid Water Christopher Adriaanse,† Jun Cheng,† Vincent Chau,†,§ Marialore Sulpizi,†,∥ Joost VandeVondele,‡,⊥ and Michiel Sprik*,† †

Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom Physical Chemistry Institute, University of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland

‡

S Supporting Information *

ABSTRACT: The electronic states of aqueous species can mix with the extended states of the solvent if they are close in energy to the band edges of water. Using density functional theory-based molecular dynamics simulation, we show that this is the case for OH− and Cl−. The effect is, however, badly exaggerated by the generalized gradient approximation leading to systematic underestimation of redox potentials and spurious nonlinearity in the solvent reorganization. Drawing a parallel to charged defects in wide gap solid oxides, we conclude that misalignment of the valence band of water is the main source of error turning the redox levels of OH− and Cl− in resonant impurity states. On the other hand, the accuracy of energies of levels corresponding to strongly negative redox potentials is acceptable. We therefore predict that mixing of the vertical attachment level of CO2 and the unoccupied states of water is a real effect. SECTION: Liquids; Chemical and Dynamical Processes in Solution

V

a solute is now possible.2−4 What these measurements reveal is that vertical IP’s of small anions such as OH− and Cl− are about 3 eV larger than the adiabatic IP’s as was already anticipated by Delahay.13,14 This reduces the separation between the vertical detachment levels (minus the IP) of these solutes and the VBM of water to less than 1 eV. The close proximity to the VBM suggests that hybridization with the extended occupied states of the solvent is after all a possibility. Similarly, the electronic states of species with a strongly negative redox potential such as CO2 could be close enough to the CBM to mix with the empty states of water. What would be the effect of such a hybridization on redox potentials? This question can in principle be answered by allatom methods, which treat solute and solvent at the same level of theory. We have developed such a method combining density functional theory-based molecular dynamics (DFTMD) and free energy perturbation (FEP).15−19 The result for OH•/ OH− reduction is summarized in Figure 1. The errors are substantial. The BLYP approximation20,21 to the redox potential is 0.6 V too small (1.3 instead of 1.9 V). The vertical IP is underestimated by 2.7 eV (7.5 vs 9.2 eV). What is even more worrying is that the same generalized gradient approximation (GGA) functionals give much more accurate results when implemented with implicit solvent methods. The reaction field estimate for the redox potential of ref 10 agrees

iewed from an electronic structure perspective, liquid water is an oxide with a band gap of 8.7 eV.1 To accommodate such a large energy gap, the valence band maximum (VBM) must lie very deep. The position of the VBM can be estimated from the onset of photoelectron emission. Adding the energy gap gives the conduction band minimum (CBM). With these definitions, the photoemission spectroscopy (PES) experiments by the Winter/Faubel group2−4 place the VBM at −9.9 and the CBM at −1.2 eV below vacuum. Represented as electrode potentials relative to the standard hydrogen electrode (SHE)5 this corresponds to a VBM at 5.5 V and a CBM at −3.2 V. These are extreme potentials for aqueous redox chemistry. Taking the OH radical as an example, the standard potential vs SHE for this highly oxidative species is +1.9 V, well below the potential of the VBM.6 At the negative end of the SHE scale, the margin is somewhat smaller. Still the potential of a redox inert species such as CO2 (−1.8 V) is separated from the CBM by more than 1 V. The band edges of the solvent seem to be out of the way. Indeed experimental redox potentials can be reproduced with impressive accuracy representing water by either a dielectric continuum (implicit solvent models)7−10 or molecular charge distributions (explicit solvent models).11,12 It may be necessary to include a small number of coordinated water molecules in the quantum calculation, but the band structure of water can be ignored.4,8 Redox potentials are adiabatic ionization potentials (IPs) providing only limited information on the electronic states involved. Vertical IPs are a more direct probe. Thanks to recent technical advances, accurate determination of the vertical IP of © 2012 American Chemical Society

Received: September 27, 2012 Accepted: November 6, 2012 Published: November 6, 2012 3411

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however, far more demanding because of the sampling required for the estimation of the effect of the thermal fluctuations of the liquid environment. Aqueous chemistry also has, however, a distinct advantage over semiconductor physics. The SHE is, from a DFT perspective, much better behaved than the VBM used as an energy reference in defect calculations.23−25 The ambiguity in the position of the VBM is an obvious source of uncertainty when charge transition levels are specified relative to the VBM.25 The SHE eliminates this problem. The H+(aq) to H2(g) reduction can be formally separated in a transfer of the proton to the gas-phase where neutralization takes place (see below). Proton solvation is closed shell chemistry and should be much less sensitive to the band gap error. The formal basis of our method is Trasatti’s theory of absolute electrode potentials.5,19 The standard potential versus SHE for reduction of an aqueous species X to X− is expressed as the sum of an electronic and ionic work function

Figure 1. Energy levels of the OH•/OH− couple aligned with the band edges of liquid water. The positions of the energy levels have been computed from total energy differences and referred to the SHE using the molecular dynamics hydrogen electrode in the version of ref 18. The BLYP calculation of the vertical IP of OH−, electron affinity (EA) of OH•, and the OH•/OH− reduction potential (U°) are taken from ref 15. The VBM and CBM of liquid water are from ref 19. For the source of the experimental data, see text.

g,◦ ◦ −(she) = AIP − + W + − μ + e0U X/X X H H

(1)

−

AIPX− is the adiabatic IP of X (aq), and WH+ is the workfunction of the aqueous proton (H+(aq)). μg,H°+ is the standard chemical potential of the gas-phase proton (H+(g)) obtained form the free energy of the reaction 1/2H2(g) → H+ (g) + e−(vac). e0 is the elementary charge (microscopic energy units are used). The last two terms of eq 1 add up to the negative of the absolute value of the potential of the SHE:5 U°H+/H2 (abs) = (μg,H°+ − WH+)/e0 = 4.44 V. The adiabatic IP in eq 1 is a f ree energy difference between oxidation states, which is not directly accessible in electronic structure calculation. However, free energy differences can be related to total energy differences using coupling integral methods. A fictitious mapping Hamiltonian /η is constructed consisting of a linear combination /η = η /O + (1 − η)/R of the atomic Hamiltonian /O of the oxidized state (X•) and /R of the reduced state (X−). The coupling parameter connecting R and O takes the values 0 ≤ η ≤ 1. The electronic workfunction in eq 1 is obtained from the thermodynamic integral

with experiment within 0.1 V. Also, the vertical IP can be accounted for in a semiquantitative way by a Born model for excited states parametrized by fitting the cavity radius to the solvation free energy14,22 The unfavorable comparison to implicit solvent methods suggests that coupling to the solvent valence band in all-atom GGA calculations is real but spurious. To understand what could be going wrong, we found it helpful to draw a parallel between electroactive aqueous solutes and charged defects in solid oxides.23−27 This isomorphism is particularly compelling for the OH•/OH− couple because these two species are ionization products of water. Viewed as a defect in water, the OH• radical becomes a hydrogen vacancy (V0H in the Kröger− Vink notation). Its negative charge state V−H is the solid state equivalent of OH−(aq) and can be interpreted as a proton vacancy.28 The problem of accurate calculation of formation energies of charged defects has occupied the computational solid state community for over two decades (for recent reviews, see refs 23−25). Defects can change charge depending on the position of the Fermi level. Estimates where these transitions take place can be more than an electronvolt in error. These discrepancies are strongly correlated with the error in the band gap29 or more precisely the band edges25 of the solid. Our calculations for OH• reduction (Figure 1) show similar symptoms. BLYP lifts the VBM 3.5 eV above the experimental PES estimate. The impression that the rising VBM is pushing up the ionization levels of OH− is hard to avoid. In fact, we will argue that the −IP level of OH− suffers the fate of a resonant impurity level and is forced into the energy gap of water30,31 Note that the agreement with experiment for the empty levels is much better. We should point out, however, that the position of the electron attachment level of OH is not known experimentally. The −EA level in the figure is an extrapolation from the known values of the −IP and adiabatic level −e0U°. Assuming linear response, the −EA level has been placed symmetrically on the opposite side of the adiabatic level. The view taken in this paper is that all-atom computations of IPs in liquids and solids have much in common, including the DFT-related problems. The calculations in solution are,

AIPX− =

∫0

1

dη⟨ΔE X−⟩η + e0V0

(2)

−

where ΔEX− is the vertical IP of X computed from total energy differences. The brackets denote a thermal average over the canonical ensemble defined by /η. The ionization is carried out in a DFTMD cell under full three-dimensional (3D) periodic boundary conditions. This introduces a bias in the reference of the electrostatic potential known in solid state physics as the band alignment problem. This bias is represented by the V0 term in eq 2. The idea behind the SHE reference is that the bias V0 in eq 2 is canceled by an opposite shift in the proton work function. WH+ is estimated from a thermodynamic integral similar to eq 2 WH+ =

∫0

1

dη⟨ΔE H3O+⟩η − ΔEzp − e0V0

(3)

where ΔEH3O+ is the energy for vertical deprotonation of a hydronium ion (H3O+). ΔEzp is a zero point motion correction. The modeling of desolvation of H+ by deprotonation of H3O+ is an approximation originally introduced for the calculation of acidity constants. The implementation and justification of this scheme is outlined in detail in refs 17 and 19. Shifts in the 3412

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error in U° is approximately linear in the potential. The discrepancy for the most electro negative species, the Cl radical, is as large as −1 V, while BLYP gets it practically right for CO2. Our interpretation of this result is that details of the individual species, such as hydrogen bonding, are evidently not very important. What matters most is how far away the redox level is from the VBM of water. In default of experimental estimates of the vertical electron affinity of the OH radical, we assumed the response of the solvent to be linear, which allowed us to complete the experimental level diagram in Figure 1 by setting the unknown reorganization energy λR in the reduced state equal to the reorganization energy λO in the oxidized state (for a definition of these quantities, see the Supporting Information). This is not what we observed for the reorganization in the BLYP system which showed a pronounced nonlinearity with a λR/λO ratio of approximately 3. A similar asymmetry was found for Cl, as can be seen in Figure 2. Symmetry is approximately restored at zero potential: λR/λO ≈ 1 for O2. Going toward negative potential reverses the asymmetry with λR/λO ≈ 1/3 for CO2. We already saw that HSE06 brings the error in U°OH/OH− down. Also, the asymmetry is effectively removed (see Figure 1). Nonlinear solvent response at positive potentials would therefore appear to be another manifestation of the misalignment of the VBM.33 However, consider now the asymmetry at the negative potential end. Here it is λR that is anomalously small. The reason is that the CBM sets an upper limit to the energy of the vertical attachment level of the oxidized species (see Figure S1). However, our CBM and redox level of CO2 are in fair agreement with experiment, suggesting that, in this case, interactions with the extended states of water are a real effect. If we could explain how interaction with the band edges is capable of inducing nonlinear solvent relaxation, the case for making this the cause of the trouble with the GGA would be considerably stronger. In a first attempt to find an answer to this question, we consider what happens if the −IP level of the solute would lie below the VBM of water. There is reason to believe that this odd condition applies to the GGA system. Both continuum representations of the solvent14,22 and QM/ MM models2 can be made to reproduce the experimental IP of OH− with a reasonable parametrization, suggesting that electrostatic interactions can account in essence for the IP. However, BLYP places the VBM more than 3 eV higher. Switching on the hybridization evidently has a drastic effect. Resonant impurity levels in semiconductors are described by the Greens function theory of Haldane and Anderson.30,31 According to this theory, the hybridization of such states with the (occupied) extended states of the solid has the tendency to insert a level in the gap just above the VBM. Our conjecture is therefore that this is why the vertical detachment level of OH− seems to move up together with the VBM of water. The hybridization is accompanied by charge transfer. This could be an explanation of why ions miss charge in aqueous solution, as has been observed by DFTMD simulation.34 Consistent with this, we found that the BLYP OH radical carries a small negative Mulliken charge of −0.14, which is reduced to −0.02 in the SHE approximation. The Anderson model has been used to rationalize charge self-regulation of transition metal impurities in semiconductors.26 This is the phenomenon that a change of oxidation state appears to have little effect on the charge of the ion. Application to (pseudo) halides in water is at this stage speculative and will need to be backed up by an appropriate parametrization of the

electrostatic reference in periodic cells are strongly dependent on composition. By using the same V0 in eqs 2 and 3, we have assumed that V0 is dominated by the solvent.17 Only when this condition is satisfied is the electrostatic zero in the “half” cell for ionization aligned with the zero in the half cell for deprotonation, and we can ignore V0 terms in the calculation of redox potentials versus SHE (eq 1). The lower limit (η = 0) for the integration in eq 2 is the vertical IP of X− again offset by e0V0. The upper limit (η = 1) is the similarly biased vertical electron affinity (EA) of X•. The uncertainty in the reference is removed by transforming to the SHE scale. The procedure is the same as that for the adiabatic IP (eq 1). Substituting eq 3, we can write the vertical IP as IPX−(she) = ⟨ΔE X−⟩η = 0 + − μHg +,◦

∫0

1

dη⟨ΔE H3O+⟩η − ΔEzp (4)

and similarly for EAX•(she). These are the “working” expressions for the computation of the vertical levels in Figure 1. In the linear response regime, the integral of eq 2 is determined by its end point values.16,17 U°x/x− (she) can be approximated by the mean of IPX−(she) and EAX•(she). The brief outline of the method presented above is meant as preparation for the discussion of the key result in Figure 1, the misalignment of the VBM of the solvent. The VBM has been computed from vertical ionization of a DFTMD model of pure liquid water following the same procedure as applied for computation of the vertical IP of the solute (eq 4). The point we want to make is that the band gap error of BLYP raises the VBM by 3.5 eV relative to experiment and that this effect is responsible for the underestimation of the IP of OH−. Two further observations support this mechanism. The first is the effect of switching to a functional with a fraction of exact exchange. Comparing the HSE0632 and BLYP result in Figure 1, we see that HSE06 has lowered the VBM by 1.3 eV pulling the vertical ionization level of OH− along. This increases U° by 0.4 to 1.7 V, i.e., 0.2 V short of the experimental value. However, the HSE06 estimate of the VBM makes it only half way, still leaving a gap of 2 eV with experiment. A second indication of the critical importance of the water band structure is the gain in accuracy for more negative potentials. This effect is displayed in Figure 2 for a set of small aqueous species. The series consists of Cl, OH, HS, HO2, O2, and CO2 listed in decreasing order of oxidative power. The

Figure 2. The error in the redox potential U° vs SHE computed using BLYP (blue circles) plotted against the experimental value for six small aqueous species. The blue line is a linear fit. The red squares give the ratio of the reorganization energies λR and λO (see Figure 1). 3413

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HECToR, the UK’s high-end computing resource funded by the Research Councils.

Anderson model. Then there is of course the question of whether most of this is merely an artifact of the delocalization error in the GGA.29 To investigate how relevant this is for real systems, we must first eliminate the error in the position of the VBM, which may require many body perturbation methods (GW). Such methods have already been employed with success for the modeling of defects in solids.27 On the other hand, we found that the uncertainties in DFT calculations of energies at negative SHE potentials are significantly smaller. Our observation of interaction between the electronic states of CO2 and the empty states in the CBM of water seems therefore to be on firmer ground. Unification of electrochemistry and electronic spectroscopy measurements is an old ideal of physical chemistry.13,14,22 Implicit solvent models for vertical electronic excitations are currently an active topic for research, and considerable progress has been made7,35,36 (see also ref 4). The advantage of these and other embedding methods is that the quantum part can be treated using methods going beyond DFT, such as correlated wave function methods.36 The parallel to defects in solid oxides outlined in this paper, however, has convinced us that for highly redox active species, it may be necessary to include the full set of extended electronic states of the solvent requiring all-atom methods.

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(1) Bernas, A.; Ferradini, C.; Jay-Gerin, J.-P. On the Electronic Structure of Liquid Water: Facts and Reflections. Chem. Phys. 1997, 222, 151−160. (2) Winter, B.; Faubel, M.; Hertel, I. V.; Pettenkofer, C.; Bradforth, S. E.; Jagoda-Cwiklik, B.; Cwiklik, L.; Jungwirth, P. Electron Binding Energies of Hydrated H3O+ and OH−: Photoelectron Spectroscopy of Aqueous Acid and Base Solutions Combined with Electronic Structure Calculations. J. Am. Chem. Soc. 2006, 128, 3864−3865. (3) Seidel, R.; Thümer, S.; Winter., B. Photoelectron Spectroscopy Meets Aqueous Solutions: Studies From a Vacuum Liquid Microjet. J. Phys. Chem. Lett. 2011, 2, 633−641. (4) Seidel, R.; Thürmer, S.; Moens, J.; Geerlings, P.; Blumberger, J.; Winter, B. Valence Photoemission Spectra of Aqueous Fe2+/3+ and [Fe(CN)6]4−/3− and their Interpretation by DFT Calculations. J. Phys. Chem. B 2011, 115, 11671−11677. (5) Trasatti, S. The Absolute Electrode Potential: An Explanatory Note. Pure Appl. Chem. 1986, 58, 955−966. (6) Stanbury, D. M. Reduction Potentials Involving Inorganic Free Radicals in Aqueous Solution. Adv. Inorg. Chem. 1989, 33, 69−138. (7) Tomasi, J.; Mennucci, B.; Cammi, R. Quantum Mechanical Continuum Solvation Models. Chem. Rev. 2005, 105, 2999−3093. (8) Jaque, P.; V., M. A.; Cramer, C. J.; Truhlar, D. Computational Electrochemistry: The Aqueous Ru3+|Ru2+ Reduction Potential. J. Phys. Chem. C 2007, 111, 5783−5799. (9) Cramer, C. J.; Truhlar, D. G. A Universal Approach to Solvation Modelling. Acc. Chem. Res. 2008, 41, 760−768. (10) Jinnouchi, R.; Anderson, A. B. Aqueous and Surface Redox Potentials from Self-Consistently Determined Gibbs Energies. J. Phys. Chem. C 2008, 112, 8747−8750. (11) Zeng, X.; Hu, H.; Hu, X.; Cohen, A. J.; Yang, W. Ab Initio Quantum Mechanical/Molecular Mechanical Simulation of Electron Transfer Rrocess: Fractional Electron Approach. J. Chem. Phys. 2008, 128, 124510. (12) Wang, L.-P.; Van Voorhis, T. A Polarizable QM/MM Explicit Solvent Model for Computational Electrochemistry in Water. J. Chem. Theor. Comput. 2012, 8, 610−617. (13) Von Burg, K.; Delahay, P. Photo Emission Spectroscopy of Inorganic Anions in Aqueous Solution. Chem. Phys. Lett. 1981, 78, 287−290. (14) Delahay, P. Photoelectron Emission Spectroscopy of Aqueous Solutions. Acc. Chem. Res. 1982, 15, 40−45. (15) Adriaanse, C.; Sulpizi, M.; VandeVondele, J.; Sprik, M. The Electron Attachment Energy of the Aqueous Hydroxyl Radical Predicted From the Detachment Energy of the Aqueous Hydroxide Anion. J. Am. Chem. Soc. 2009, 131, 6046−6047. (16) Cheng, J.; Sulpizi, M.; Sprik, M. Redox Potentials and pKa for Benzoquinone from Density Functional Theory Based Molecular Dynamics. J. Chem. Phys. 2009, 131, 154504. (17) Costanzo, F.; Della Valle, R. G.; Sulpizi, M.; Sprik, M. The Oxidation of Tyrosine and Tryptophan Studied by a Molecular Dynamics Normal Hydrogen Electrode. J. Chem. Phys. 2011, 134, 244508. (18) Cheng, J.; Sulpizi, M.; Joost VandeVondele, J.; Sprik, M. Hole Localization and Thermochemistry of Oxidative Dehydrogenation of Aqueous Rutile TiO2(110). ChemCatChem 2012, 4, 636−640. (19) Cheng, J.; Sprik, M. Alignment of Electronic Energy Levels at Electrochemical Interfaces. Phys. Chem. Chem. Phys. 2012, 14, 11245− 11267. (20) Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behaviour. Phys. Rev. A 1988, 38, 3098−3100. (21) Lee, C.; Yang, W.; Parr, R. Development of the Colle−Salvetti Correlation-Energy Formula into a Functional of Electron-Density. Phys. Rev. B 1988, 37, 785−789.

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METHOD The DFTMD model system for liquid water consists of 32 water molecules in a periodic cubic cell of length 9.86 Å. The model for OH−(aq) contains one proton less.18 The model systems for the other solutes in Figure 2 are generated by replacing one water molecule by the solute. The valence orbitals are expanded in a TZV2P Gaussian basis. Core electrons are represented by pseudo potentials. The DFTMD simulations were carried out using the CP2K package (http:// www.cp2k.org). HSE06 has been implemented using the ADMM method.37 No finite system size corrections have been applied. Further technical details can be found in the Supporting Information.

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ASSOCIATED CONTENT

S Supporting Information *

Tables with the data for the figures and an error analysis. This material is available free of charge via the Internet at http:// pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Addresses §

University of Hong Kong. Johannes Gutenberg University Mainz. ⊥ Department of Materials, ETH Zurich. ∥

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The Engineering and Physical Sciences Research Council (EPSRC) is acknowledged for financial support for C.A., J.C., and M.S. J.C. thanks Emmanuel College at Cambridge for a research fellowship. Part of the calculations have been performed using the UKCP share of computer time on 3414

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(22) Coe, J. V.; Earhart, A. D.; Cohen, M. H.; Hoffman, G. J.; Sarkas, H. W.; Bowen, K. H. Using Cluster Studies to Approach the Electronic Structure of Bulk Water: Reassessing the Vacuum Level, Conduction Band Edge, and Band Gap of Water. J. Chem. Phys. 1997, 107, 6023− 6031. (23) VandeWalle, C. G.; Janotti, A. Advances in Electronic Structure Methods for Defects and Impurities in Solids. Phys. Status Solidi B 2011, 248, 19−27. (24) Deak, P.; Gali, A.; Aradi, B.; Frauenheim, T. Accurate Gap Levels and Their Role in the Reliability of Other Calculated Defect Properties. Phys. Status Solidi B 2011, 248, 790−798. (25) Alkauskas, A.; Broqvist, P.; Pasquarello, A. Defect Levels through Hybrid Density Functionals: Insights and Applications. Phys. Status Solidi B 2011, 248, 775−789. (26) Raebiger, H.; Lany, S.; Zunger, A. Charge Self-Regulation on Changing the Oxidation State of Transition Metals in Insulators. Nature 2008, 453, 763−766. (27) Rinke, P.; Janottti, A.; Scheffler, M.; Van de Walle, C. G. Defect Formation Energies without the Band-Gap Problem: Combining Density-Functional Theory and the GW Approach for the Silicon SelfInterstitial. Phys. Rev. Lett. 2009, 102, 026402. (28) Maier, J. Acid−Base Centers and Acid−Base Scales in Ionic Solids. Chem.Eur. J. 2001, 7, 4763−4770. (29) Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction. Phys. Rev. Lett. 2008, 100, 146401. (30) Haldane, F. D. M.; Anderson, P. W. Simple Model of Multiple Charge States of Transition-Metal Impurities in Semiconductors. Phys. Rev. B 1976, 13, 2553−2559. (31) Davydov, S. Y.; Troshin, S. V. Adsorption on Metals and Semiconductors: Anderson−Newns and Haldane−Anderson Models. Phys. Solid State 2007, 49, 1583−1588. (32) Krukau, A. V.; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106. (33) In ref 15 we attributed this to a non-linear effect in the reorganization of the hydrogen-bonding. We no longer support this explanation. (34) Dal Perraro, M.; Raugei, S.; Carloni, P.; Klein, M. L. SoluteSolvent Charge Transfer in Aqueous Solution. ChemPhysChem 2005, 6, 1715−1718. (35) Chipman, D. M. Vertical Electronic Excitation with a Dielectric Continuum Model of Solvation Including Volume Polarization. I. Theory. J. Chem. Phys. 2009, 131, 014103. (36) Fukuda, R.; Ehara, M.; Nakatsuji, H.; Cammi, R. Nonequilibrium Solvation for Vertical Photoemission and Absorption Processes Using the Symmetry-Adapted Cluster-Configuration Method in the Polarizable Continuum Method. J. Chem. Phys. 2011, 134, 104109. (37) Guidon, M.; Hutter, J.; VandeVondele, J. Auxiliary Density Matrix Methods for Hartree−Fock Exchange Correlations. J. Chem. Theor. Comput. 2010, 6, 2348−2364.

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